 #1
Buzz Bloom
Gold Member
 2,211
 365
Summary:

I am trying to understand the use of the Schwarzschild metric in measuring the "proper" distance between two nonmoving concentric spherical shells which are also both concentric with the event horizon sphere which of course has the Schwarzschild radius.
If what I describe below is correct, I hope someone will confirm this for me. If it is is incorrect, I hope someone with explain my error to me.
Main Question or Discussion Point
For the purpose of this thread the metric is
I assume that the two spherical shells are stationary. Therefore
The proper radial distance D between the r_{1} shell and the r_{2} shell is:
My purpose in studying this problem is that I am interested in estimating the error if I use Newtonian gravity equations for orbital motion rather than the more accurate (but more difficult to use) Schwarzschild math.
ds^{2} =  (1r_{s}/r) c^{2} dt^{2} + dr^{2} / (1r_{s}/r)
wherer_{s} = 2GM/c^{2}.
(I modified the above fromI assume that the two spherical shells are stationary. Therefore
dt = 0.
The r coordinate for the radii of the two shells satisfy the relationships:A_{1} = 4 π (r_{1})^{2}
is the surface area of one spherical shell, andA_{2} = 4 π (r_{2})^{2}
is the area of the other spherical shell.The proper radial distance D between the r_{1} shell and the r_{2} shell is:
D = ∫_{r1}^{r2} (1/(1r/r_{s}))^{1/2} dr.
My purpose in studying this problem is that I am interested in estimating the error if I use Newtonian gravity equations for orbital motion rather than the more accurate (but more difficult to use) Schwarzschild math.